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Required area

$$ = \int\limits_0^1 {(2x - {x^2})dx} $$

$$ = \left[ {2.{{{x^2}} \over 2} - {{{x^3}} \over 3}} \right]_0^1$$

$$ = \left[ {{x^2} - {{{x^3}} \over 3}} \right]_0^1$$

$$ = \left( {1 - {1 \over 3}} \right)$$

$$ = {2 \over 3}$$

Let, the equation of line passes through the origin is y = mx.

This line and x = 1 line cut at point B.

y = m . 1 = m

$$\therefore$$ Point B = (1, m)

Area of $$\Delta OAB = {1 \over 2} \times OA \times AB$$

$$ = {1 \over 2} \times 1 \times m$$

Given that straight line cut the area two equal parts.

$$\therefore$$ 2 $$\times$$ Area of $$\Delta OAB = {2 \over 3}$$

$$ \Rightarrow 2 \times {1 \over 2} \times 1 \times m = {2 \over 3}$$

$$ \Rightarrow m = {2 \over 3}$$

$$\therefore$$ Equation of straight line,

$$y = {2 \over 3}x$$

19:T52e,

WB JEE 2021 Mathematics - Application of Integration Question 15 English Explanation

Application Of Integration

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